3D parallel frequency-domain acoustic wave modeling:
application to full-waveform tomography.
Stéphane Operto
UMR Géosciences Azur – CNRS, FRANCE
Email: operto@geoazur.obs-vlfr.fr
Thanks to the tremendous increase of the computation power offered by large-scale
distributed-memory platforms, 3D acoustic full-waveform tomography (FWT) is today
feasible. FWT refers to the seismic imaging methods based on the full solution of the twoway wave equation for the forward problem and on numerical optimization for the inverse
problem. The resolution of the forward problem is the most computationally intensive task in
the inversion algorithm. In the frame of seismic tomography, it must be solved a huge number
of times, this number being proportional to the number of sources (several thousands in 3D)
and to the number of inversion iterations. Designing efficient massively parallel modeling
engine is therefore a necessary condition to make 3D FWT feasible.
In 2D, frequency-domain FWT has been shown to be attractive for several reasons. First, it
provides the most natural framework to design a multiresolution imaging by proceeding by
successive inversions of increasing frequencies. This helps to mitigate the non linearity of the
inverse problem. Second, a resolution analysis of FWT shows that only few frequencies need
to be inverted to build an image of the medium provided that the acquisition geometry
samples a sufficiently broad range of apertures. Third, these few frequency components are
efficiently computed for a large number of sources with a direct solver once the matrix
resulting from the discretization of the forward problem was LU factorized.
The extension to 3D of the direct-solver modeling approach is not so attractive because of the
memory and time complexity of direct solvers which become prohibitive for large 3D
problems. I will first discuss the size of the imaging problems that can be tackled with such
approach. Second, I will introduce a domain decomposition method for frequency-domain
wave modeling whose aim is to overcome the memory requirement of the direct approach.
This domain decomposition method is based on a hybrid direct/iterative solver and on the
Schur complement approach. The direct solver is applied in sequential to each sub-domain.
The iterative solver based on the GMRES algorithm is used to solve the Schur complement
system whose solution gives the wavefield at the interface grid points between subdomains.
The preconditioning of the Schur complement system is based on an additive Schwarz
approach and is provided by the sum of the local Schur complements. I will show how the
hybrid approach allows to decrease the memory cost of the frequency-domain modeling
compared to the direct approach at the partial expense of the CPU-time efficiency for
multiple-source simulations. Indeed, when the number of sub-domains increases, the memory
requirement of the hybrid solver and the CPU time of each elementary task on each processor
decrease but the number of iterations of GMRES increases due to the degradation of the
preconditioner. I will illustrate this antagonistic behavior of the hybrid solver with simulations
performed on 2000 processors. To maintain tractable computational time, the hybrid solver
requires data reduction based on source stacking. I will illustrate the footprint of this data
reduction on the tomographic reconstruction.
A third strategy recently proposed in the literature solves the forward problem in the time
domain while leaving the inversion in the frequency domain. Complexity analysis suggests
that it’s the most efficient approach. I will present parallel time-domain simulations to
compare the conclusions of theoretical complexity analysis with numerical experiments.
In the end, I will present some synthetic experiments to illustrate the resolution power of 3D
FWT and the footprint of coarse acquisition geometries on the imaging.